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This article presents a novel methodology for analyzing the resilience of an active distribution system (ADS) integrated with an urban gas network (UGN). It demonstrates that the secure adoption of gas turbines with optimal capacity and allocation can enhance the resilience of the ADS during high-impact, low-probability (HILP) events. A two-level tri-layer resilience problem is formulated to minimize load shedding as the resilience index during post-event outages. The challenge of unpredictability is addressed by an adaptive distributionally robust optimization strategy based on multi-cut Benders decomposition. The uncertainties of HILP events are modeled by different moment-based probability distributions. In this regard, considering the nature of each uncertain variable, a different probabilistic method is utilized. For instance, to account for the influence of power generated from renewable energy sources on the decision-making process, a diurnal version of the long-term short-term memory network is developed to forecast day-ahead weather. In comparison with standard LSTM, the proposed approach reduces the mean absolute error and root mean squared error by approximately 47% and 71% for wind speed, as well as 76% and 77% for solar irradiance network. Finally, the optimal operating framework for improving power grid resilience is validated using the IEEE 33-bus ADS and 7-node UGN.